<p><it>Abstract—</it>An enumeration of star graphs is given which has many useful properties. For example an arbitrary prefix or suffix is connected; indeed the diameter is <math><tmath>$O(n)$</tmath></math>. As a consequence, there is an <math><tmath>$O(n)$</tmath></math> interval broadcast algorithm. Prefixes which have <math><tmath>$t(n-1)!$</tmath></math> vertices for some <math><tmath>$t$</tmath></math> are especially well-behaved. The topology of, embeddings in, and algorithms for these graphs are considered, making use of the enumeration.</p><p><it>Index Terms—</it>Interconnection network, star graph, incomplete star graph, interval broadcast, Hamiltonian cycles.</p>